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List of Publications with M. Wolf: Constraint and super Yang-Mills equations on the deformed 4|16 superspace Rh , JHEP 0403 (2004) 048 [hep-th/0401147]. with A.D.Popov: On supertwistors, the Penrose-Ward transform and N=4 super Yang-Mills theory, to appear in ATMP [hep-th/0405123].
The topological B-model on fattened complex manifolds and subsectors of N=4 self-dual Yang-Mills theory, JHEP 0501 (2005) 042 [hep-th/0410292]. with A.D. Popov and M. Wolf: The topological B-model on a mini-supertwistor space and supersymmetric Bogomolny monopole equations, JHEP 0510 (2005) 058 [hep-th/0505161]. with M. Ihl: Drinfeld twisted supersymmetry and non-anticommutative superspace, JHEP 0601 (2006) 065 [hep-th/0506057].
On the mini-superambitwistor space and N=8 super Yang-Mills theory, submitted to CMP, hep-th/0508137. with O. Lechtenfeld: Matrix models and D-branes in twistor string theory, submitted to JHEP, hep-th/0511130.
Outline
Introduction Extensions of Spacetime (SUSY, NAC, Twistors) Gauge Theories String Theory and D-Branes Drinfeld Twists and Non-Anticommutativity The Mini-Superambitwistorspace Matrix Models and D-Branes in Twistor String Theory Conclusions
Extensions of Spacetime SSttaannddaarrdd MMooddeell: very successful! but: evidently not down to arbitrary distances (e.g. gravity, Landau pole) Extensions of Spacetime: Supersymmetry: add fermionic dimensions
Noncommutativity:
Both: Non-anticommutativity:
Twistors: add celestial spheres:
Supersymmetry
acts as translations in fermionic dimensions hierarchy problem solved SUSY is broken unification of gauge couplings (e.g. "½" spectrum) candidate for dark matter Higgs should be found soon non-renormalization theorems SUGRA non-renormalizable upon localization, SUGRA appears possibly all particles in one multiplet leads to “nicer“ string theories nice new mathematical structu res Non-commutativity in various situations in string theory and M-theory
functions become operators on a Hilbert space QFTs get stringy featu res, but IR/UV mixing Non-anticommutativity
from graviphoton background in string theory
missing SUSY introduces problems: renormalizability hard to prove chiral rings and WT-Identities missing calculations much more involved
Summing up the situation ...... but we will see that we can do better! Twistors add to every point in space its “celestial sphere“ with a natural complex structure:
base: fibres: overlap: base: fibres:
Twistor correspondence
Complexify the picture:
sphere point point null-plane Penrose-Ward transform: holomorphicity Integrability solutions to
o n twistor side on null- spaces Yang-Mills Gauge Theories I equations: , , Field strength in spinor notation:
Yang-Mills self-duality SUSY ad libitum Yang-Mills-Higgs BPS
Na hm and ADHM equations Gauge Theories II topological gauge theory: Chern-SiSimons theory
Complex category: replace with holomorphic Chern-Simons theory:
need Calabi-Yau manifolds
String Theory
String theory: open closed theory of embeddings of 2d manifolds into spacetime inevitably, D-branes stack of n D-branes, ends appear carry gauge theory labels: N=1, d=10: U(n) SYM N=2, d=4: U(n) SDYM N=2, d=6 (top.): GL(n,C) hCS Drinfeld Twist
Drinfeld Twist
Model for: Consider the super Poincaré algebra as a Hopf algebra with coproduct Twist the coproduct:
On the representation space (algebra of functions): yielding without destroying SUSY (succesfully intr oduced before) Drinfeld Twist: Advantages
Twisted supersymmetry and chirality preserved! ✔ Representation content identical ✔ Vacuum energy 0 (in agreement with literature) ✔ SUSY chiral rings can be introduced ✔ Twisted Ward-Takahashi identities ✔ Non-renormalization? Naturalness argument looks promising. (different suggestion from one-loop calculations) Mini-Superambitwistors
Twistor String Theory
Calabi- Yau Twistor
Twistor String Theory
String theory on supertwistor space yields new string/gauge duality
top. B-model on super CY
holomorphic Chern-Simons
holomorphicity integrability solutions to
on super CY on null- spaces Yang-Mills Supertwistorspaces four dimensions: N=4 SUSY SDYM N=3,4 super YM three dimensions: N=8 Bogomolny evidently missing: N=8 super YM Construction of mini-superambitwistor space
The Mini-Superambitwistor Space
What is a quadric in ? abstract definition: more explicitly:
x
fi bration not vector bundle Mini-Superambitwistor geometry
Interpretation via spaces of oriented lines:
Further remarks
after clarifying some technicalities: PW works! (even for a bosonic truncation) holomorphicity integrability Yang-Mills- on L4|6 on null-lines Higgs
What about topological B-model and twistor strings? Inconclusive: degeneracy cycles ( CY condition) OK unclear, how to define holomorphic Chern-Simons
Mirror conjecture: Twistor Matrix Models
Twistor Matrix Model Matrix Models from dimensional reduction Twistor side Spacetime side
Penrose-Ward transform preserved: hCS MQM SDYM MM Alternative Matrix Models Introduce noncommutativity:
fields become operators/matrices, derivatives become commutators, integrals bec ome traces, First Results
Matrix Models from dimensional reduction SDYM MM describes D(-1)-branes in N=2 ST hCS MM describes D1-branes in top. B-model fermionic dimensions smeared out
Matrix Models from noncommutativity SDYM MM describes D3-branes in N=2 ST in a B-field background hCS MM describes D5-branes in top. B-model in a B-field background
use twistor methods to construct solutions Further Results: ADHM eqns. D(-1) Two equivalent descriptions: D3 via D3-branes: D(-1) is an instanton:
solution to SDYM (ch2≠0) via D(-1)-brane: all different strings build up to ADHM eqns. (≈ SDYM MM) in topological theory: D5 D1 via D5-branes: D1 are solutions to hCS via D1-branes: extension of hCS MM Further Results: Nahm eqns.
D3 Two equivalent descriptions: via D3-branes: D1 is a monopole: D1 solution to Bogomolny via D1-brane: D3 D1-D1-strings build up to Nahm eqns. (≈ 1d SDYM) in topological theory: D3 via D3-branes: D3' solutions to hCS via D3'-branes: D3 reduction of hCS to D3 ' D3' Conclusions I
Drinfeld Twisted Non-Anticommutativity
Done: Future directions: ✔ SUSY NAC deformation ✗ investigate situation ✔ consistency checks non-renormalizability ✔ representation content ✗ superconformal twist ✔ reconstruction of chiral ✗ NAC supergravity rings and WT-identities ✗ convince people to
✔ naturalness argument use our approach Conclusions II
The Mini-Superambitwistor Space
Done: Future directions: ✔ Construction of mini- ✗ define hCS theory/ superambitwistor space a topological B-model ✔ clarification of geometry ✗ substantiate mirror ✔ technicalities solved conjecture ✔ PW transform N=8 ✗ adapt recent construct.
✔ PW transform N=0 of twistor actions Conclusions III
Matrix Models & D-branes in Twistor String Theory
Done: Future directions: ✔ definition of four MMs ✗ clarify n to infinity ✔ PW transform preserved ✗ study Nahm in detail ✔ D-brane interpretation ✗ mirror symmetry and ✔ extension to ADHM T-duality ✔ further MMs for Nahm ✗ carry over topological
equations tools to IIB D-branes AAssppeeccttss ooff TTwwiissttoorr GGeeoommeettrryy aanndd SSuuppeerrssyymmmmeettrriicc FFiieelldd TThheeoorriieess wwiitthhiinn SSuuppeerrssttrriinngg TThheeoorryy
PPrroommoottiioonnssvvoorrttrragag
CChhrriissttiiaann SSäämmaannnn
IInnsstitittuutt füfürr tthheeoorretetiisscchhee PPhhyyssiikk UUnniivveerrssiittäätt HHaannnnoovveerr
3300.. JJaannuuaarr 22000066